A230547 -id:A230547 - cp's OEIS frontend

A006629 Self-convolution 4th power of A001764, which enumerates ternary trees.

1, 4, 18, 88, 455, 2448, 13566, 76912, 444015, 2601300, 15426840, 92431584, 558685348, 3402497504, 20858916870, 128618832864, 797168807855, 4963511449260, 31032552351570, 194743066471800, 1226232861415695

Offset: 0

Simon Plouffe, N. J. A. Sloane

Sum of root degrees of all noncrossing trees on nodes on a circle. - Emeric Deutsch

H. M. Finucan, Some decompositions of generalized Catalan numbers, pp. 275-293 of Combinatorial Mathematics IX. Proc. Ninth Australian Conference (Brisbane, August 1981). Ed. E. J. Billington, S. Oates-Williams and A. P. Street. Lecture Notes Math., 952. Springer-Verlag, 1982.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Andrew Howroyd, Table of n, a(n) for n = 0..200
Emanuele Munarini, Shifting Property for Riordan, Sheffer and Connection Constants Matrices, Journal of Integer Sequences, Vol. 20 (2017), Article 17.8.2.
Joris Nieuwveld, Fractions, Functions and Folding. A Novel Link between Continued Fractions, Mahler Functions and Paper Folding, Master's Thesis, arXiv:2108.11382 [math.NT], 2021.
C. H. Pah, Single polygon counting on Cayley Tree of order 3, J. Stat. Phys. 140 (2010) 198-207.
Index entries for sequences related to rooted trees

Column 2 of A092276.

Cf. A000139, A001764, A006013, A006630, A006631, A230547, A233657.

A006629:= func< n | 2*Binomial(3*n+3,n)/(n+2) >;
[A006629(n): n in [0..40]]; // G. C. Greubel, Aug 29 2025

Table[2*Binomial[3*n+3,n]/(n+2), {n,0,40}] (* G. C. Greubel, Aug 29 2025 *)

a(n)=my(m=4);binomial(3*n+m-1,n)*m/(2*n+m) /* 4th power of A001764 with offset n=0 */ \\ Paul D. Hanna, May 10 2008

def A006629(n): return 2*binomial(3*n+3,n)//(n+2)
print([A006629(n) for n in range(41)]) # G. C. Greubel, Aug 29 2025

a(n) = 2*binomial(3*n+3,n)/(n+2). - Emeric Deutsch
a(n) = (n+1) * A000139(n+1). - F. Chapoton, Feb 23 2024
G.f.: hypergeom( [ 2, 5/3, 4/3 ]; [ 3, 5/2 ]; 27*x/4 ).
G.f.: A(x) = G(x)^4 where G(x) = 1 + x*G(x)^3 = g.f. of A001764 giving a(n)=C(3n+m-1,n)*m/(2n+m) at power m=4 with offset n=0. - Paul D. Hanna, May 10 2008
G.f.: (((4*sin(arcsin((3*sqrt(3*x))/2)/3))/(sqrt(3*x))-1)^2-1)/(4*x). - Vladimir Kruchinin, Feb 17 2023
E.g.f.: hypergeom([4/3, 5/3, 2]; [1, 5/2, 3]; 27*x/4). - G. C. Greubel, Aug 29 2025

More precise definition from Paul D. Hanna, May 10 2008

A233657 a(n) = 10 * binomial(3n+10,n)/(3n+10).

Original entry on oeis.org

1, 10, 75, 510, 3325, 21252, 134550, 848250, 5340060, 33622600, 211915132, 1337675430, 8458829925, 53591180360, 340185835500, 2163581913780, 13786238414025, 88004926973250, 562763873596575, 3604713725613000, 23126371951808268, 148594788106641360

Offset: 0

Tim Fulford, Dec 14 2013

Fuss-Catalan sequence is a(n,p,r) = r*binomial(np+r,n)/(np+r), this is the case p=3, r=10.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
David Bevan, Robert Brignall, Andrew Elvey Price and Jay Pantone, A structural characterisation of Av(1324) and new bounds on its growth rate, arXiv preprint arXiv:1711.10325 [math.CO], 2017-2019.
J-C. Aval, Multivariate Fuss-Catalan Numbers, arXiv:0711.0906 [math.CO], 2007; Discrete Math., 308 (2008), 4660-4669.
Thomas A. Dowling, Catalan Numbers Chapter 7
Wojciech Mlotkowski, Fuss-Catalan Numbers in Noncommutative Probability, Docum. Mathm. 15: 939-955 (2010).
Emanuele Munarini, Shifting Property for Riordan, Sheffer and Connection Constants Matrices, Journal of Integer Sequences, Vol. 20 (2017), Article 17.8.2.

Column 5 of A092276.

Cf. A000108, A001764, A006013, A006629, A102893, A006630, A102594, A006631, A230547.

[10*Binomial(3*n+10, n)/(3*n+10): n in [0..30]];

A233657:=n->10*binomial(3*n+10,n)/(3*n+10): seq(A233657(n), n=0..20); # Wesley Ivan Hurt, Oct 10 2014

Table[10 Binomial[3 n + 10, n]/(3 n + 10), {n, 0, 30}]

PARI
```
a(n) = 10*binomial(3*n+10,n)/(3*n+10);
```

{a(n)=local(B=1); for(i=0, n, B=(1+x*B^(3/10))^10+x*O(x^n)); polcoeff(B, n)}

G.f. satisfies: B(x) = {1 + x*B(x)^(p/r)}^r, here p=3, r=10.
+2*n*(n+5)*(2*n+9)*a(n) -3*(3*n+7)*(n+3)*(3*n+8)*a(n-1)=0. - R. J. Mathar, Feb 16 2018
E.g.f.: F([10/3, 11/3, 4], [1, 11/2, 6], 27*x/4), where F is the generalized hypergeometric function. - Stefano Spezia, Oct 08 2019

A355172 The Fuss-Catalan triangle of order 2, read by rows. Related to ternary trees.

Original entry on oeis.org

1, 0, 1, 0, 1, 3, 0, 1, 5, 12, 0, 1, 7, 25, 55, 0, 1, 9, 42, 130, 273, 0, 1, 11, 63, 245, 700, 1428, 0, 1, 13, 88, 408, 1428, 3876, 7752, 0, 1, 15, 117, 627, 2565, 8379, 21945, 43263, 0, 1, 17, 150, 910, 4235, 15939, 49588, 126500, 246675

Offset: 0

Peter Luschny, Jun 25 2022

The Fuss-Catalan triangle of order m is a regular, (0, 0)-based table recursively defined as follows: Set row(0) = [1] and row(1) = [0, 1]. Now assume row(n-1) already constructed and duplicate the last element of row(n-1). Next apply the cumulative sum m times to this list to get row(n). Here m = 2. (See the Python program for a reference implementation.)

This definition also includes the Fuss-Catalan numbers A001764(n) = T(n, n), or row 3 in A355262. For m = 1 see A355173 and for m = 3 A355174. More generally, for n >= 1 all Fuss-Catalan sequences (A355262(n, k), k >= 0) are the main diagonals of the Fuss-Catalan triangles of order n - 1.

			Table T(n, k) begins:
  [0] [1]
  [1] [0, 1]
  [2] [0, 1,  3]
  [3] [0, 1,  5, 12]
  [4] [0, 1,  7, 25,  55]
  [5] [0, 1,  9, 42, 130,  273]
  [6] [0, 1, 11, 63, 245,  700, 1428]
  [7] [0, 1, 13, 88, 408, 1428, 3876, 7752]
Seen as an array reading the diagonals starting from the main diagonal:
  [0] 1, 1,  3, 12,  55,  273,  1428,   7752,   43263,  246675, ...  A001764
  [1] 0, 1,  5, 25, 130,  700,  3876,  21945,  126500,  740025, ...  A102893
  [2] 0, 1,  7, 42, 245, 1428,  8379,  49588,  296010, 1781325, ...  A102594
  [3] 0, 1,  9, 63, 408, 2565, 15939,  98670,  610740, 3786588, ...  A230547
  [4] 0, 1, 11, 88, 627, 4235, 27830, 180180, 1157013, 7396972, ...

A001764 (main diagonal), A102893 (subdiagonal), A102594 (diagonal 2), A230547 (diagonal 3), A005408 (column 2), A071355 (column 3), A006629 (row sums), A143603 (variant).

Cf. A123110 (triangle of order 0), A355173 (triangle of order 1), A355174 (triangle of order 3), A355262 (Fuss-Catalan array).

from functools import cache
from itertools import accumulate
@cache
def Trow(n: int) -> list[int]:
    if n == 0: return [1]
    if n == 1: return [0, 1]
    row = Trow(n - 1) + [Trow(n - 1)[n - 1]]
    return list(accumulate(accumulate(row)))
for n in range(9): print(Trow(n))

The general formula for the Fuss-Catalan triangles is, for m >= 0 and 0 <= k <= n:
FCT(n, k, m) = (m*(n - k) + m + 1)*(m*n + k - 1)!/((m*n + 1)!*(k - 1)!) for k > 0 and FCT(n, 0, m) = 0^n. The case considered here is T(n, k) = FCT(n, k, 2).
T(n, k) = (2*n - 2*k + 3)*(2*n + k - 1)!/((2*n + 1)!*(k - 1)!) for k > 0; T(n, 0) = 0^n.
The g.f. of row n of the FC-triangle of order m is 0^n + (x - (m + 1)*x^2) / (1 - x)^(m*n + 2), thus:
T(n, k) = [x^k] (0^n + (x - 3*x^2)/(1 - x)^(2*n + 2)).

cp's OEIS Frontend

A006629 Self-convolution 4th power of A001764, which enumerates ternary trees.

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A233657 a(n) = 10 * binomial(3n+10,n)/(3n+10).

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A355172 The Fuss-Catalan triangle of order 2, read by rows. Related to ternary trees.

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cp's OEIS Frontend

A006629 Self-convolution 4th power of A001764, which enumerates ternary trees.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Magma

Mathematica

PARI

SageMath

Formula

Extensions

A233657 a(n) = 10 * binomial(3*n+10,n)/(3*n+10).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

PARI

Formula

A355172 The Fuss-Catalan triangle of order 2, read by rows. Related to ternary trees.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Python

Formula

A233657 a(n) = 10 * binomial(3n+10,n)/(3n+10).